Table of Contents
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION-2026 FOR RECRUITMENT TO POSTS IN BS-17 UNDER THE FEDERAL GOVERNMENT
APPLIED MATHEMATICS
| TIME ALLOWED: THREE HOURS | MAXIMUM MARKS = 100 |
NOTE:
- Attempt only FIVE questions in all. ALL questions carry EQUAL marks.
- All the parts (if any) of each Question must be attempted at one place instead of at different places.
- Write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
- No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
- Extra attempt of any question or any part of the attempted question will not be considered.
- Use of Calculator is allowed.
Q. No. 1
| (a) | Prove $\nabla f(r) = \frac{f'(r)}{r}\vec{r}$, and evaluate $U$ for $\nabla U = 2r^4\vec{r}$. | (8) |
| (b) | If $\vec{A}$ and $\vec{B}$ are differentiable vector function of position $(x, y, z)$, find $\nabla \cdot (\vec{A} \times \vec{B})$. | (7) |
| (c) | If $\vec{A} = x^2yz\hat{i} – 2xz^3\hat{j} + xz^2\hat{k}$ and $\vec{B} = 2z\hat{i} + y\hat{j} – x^2\hat{k}$, find $\frac{\partial^2}{\partial x \partial y}(\vec{A} \times \vec{B})$. | (5) |
Q. No. 2
| (a) | What is the maximum range possible for a projectile fired from a cannon having muzzle velocity 1 mile/sec? What is the height reached in this case? | (10) |
| (b) | Find the radial and transverse components of the velocity of the particle moving along the curve $ax^2 + by^2 = 1$ at any time $t$, if the polar angle $\theta = ct^2$. | (10) |
Q. No. 3
| (a) | Verify that the linear differential equation
$(1 + x^2)\frac{dy}{dx} + 2xy = e^{x^2}y$
can be solved by setting $y = ve^{x^2}$. Find the equation satisfied by $v$. |
(5) |
| (b) | Explain how a first-order linear differential equation can be used to model pollution accumulation in a lake with constant inflow and outflow rates. | (5) |
| (c) | Find the solution of the following differential equation
$y”’ + 8y = 2x – 5 + 8e^{-2x}$; with the conditions
$y(0) = -5$, $y'(0) = 3$, $y”(0) = -4$
|
(10) |
Q. No. 4
| (a) | Find the general solution of the following differential equation on interval $(0, \infty)$
$16x^2y” + 16xy’ + (16x^2 – 1)y = 0$
|
(10) |
| (b) | Solve the following PDEs using transformation
$u_x + u_t + u = 0$; for $x > 0, t > 0$,
$u(0, t) = \sin(t)$; $u(x, 0) = 0$.
|
(10) |
Q. No. 5
| (a) | Model the one-dimensional wave equation. | (8) |
| (b) | A stretched string of length $L$ is lying along $x$-axis and is fixed at both ends $x = 0$ and $x = L$. Find the deflection $u(x, t)$ of the string at any time $t$, if initial displacement $u(x, 0) = f(x) = Lx – x^2$ and the initial velocity is
$u_t(x, 0) = g(x) = 4$.
|
(12) |
Q. No. 6
| (a) | Find the Fourier series of the function
$f(x) = \left(1 – \frac{|x|}{a}\right) H\left(1 – \frac{|x|}{a}\right)$.
Where $H(x)$ is the Heaviside unit step function defined by
$H(x) = \begin{cases} 1, & \text{for } x > 0 \\ 0, & \text{for } x < 0 \end{cases}$
|
(10) |
APPLIED MATHEMATICS (2026)
| (b) | Evaluate the integral $I = \int_{0}^{1} \frac{1}{1+x^4} dx$ by using:
|
(10) |
Q. No. 7
| (a) | Use the Modified Euler method to solve the IVP:
$y’ = y – x^2 + 1; \quad y(0) = 0.5$
for $0 \leq x \leq 2$ with step size $h = 0.5$. |
(10) |
| (b) | Implement appropriate technique to solve the following linear system:
$3x_1 – x_2 + x_3 = -1$
$-x_1 + 3x_2 – x_3 = 7$ $x_1 + x_2 – 3x_3 = -7$ |
(10) |
Q. No. 8
| (a) | Set up Newton’s scheme of iteration for finding the square root of a positive number $N$ and evaluate $\sqrt{14}$. | (10) |
| (b) | Find the first four iterations of the equation $f(x) = x – 0.8 – 0.2\sin x$ in the interval $\left[0, \frac{\pi}{2}\right]$ using Newton-Raphson Method. | (10) |
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